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Has anyone every investigated what happens if you restrict Wang tiles to only include those where two given sides (say north and west) have the same color? For example, are there still sets which tile the plane aperiodically but not periodically? They have a symmetric geometric interpretation if you rotate the plane a bit and use a hexagonal instead of a square lattice. For n colors, instead of 24n possible sets there are only 23n, which seems like a lot less. At first I thought that problems involving this simplified form would be solvable in polynomial time, but now I'm starting to believe they have the same complexity as full Wang tiles. I'm not really expecting an answer here, but you never know until you ask. -- RDBury ( talk) 15:24, 28 June 2021 (UTC)
0 1 2 0 1, 1 2, 2 0 2 0 1
a a a b a b -> c c b c
For the regular Polyhedra (with n sides), I've managed to work out that the faces can *always* be split into q equal connected pieces if q divides n *except* for 4 and 5 face pieces on the icosahedra. Can someone help me figure those out (or let me know that there is no such?) For example, a Dodecahedron can be covered by 6 identical 2 pentagon units, 4 identical 3 pentagon units, 3 identical 4 pentagon units and 2 identical 6 pentagon units. Naraht ( talk) 18:43, 28 June 2021 (UTC)
Mathematics desk | ||
---|---|---|
< June 27 | << May | June | Jul >> | June 29 > |
Welcome to the Wikipedia Mathematics Reference Desk Archives |
---|
The page you are currently viewing is a transcluded archive page. While you can leave answers for any questions shown below, please ask new questions on one of the current reference desk pages. |
Has anyone every investigated what happens if you restrict Wang tiles to only include those where two given sides (say north and west) have the same color? For example, are there still sets which tile the plane aperiodically but not periodically? They have a symmetric geometric interpretation if you rotate the plane a bit and use a hexagonal instead of a square lattice. For n colors, instead of 24n possible sets there are only 23n, which seems like a lot less. At first I thought that problems involving this simplified form would be solvable in polynomial time, but now I'm starting to believe they have the same complexity as full Wang tiles. I'm not really expecting an answer here, but you never know until you ask. -- RDBury ( talk) 15:24, 28 June 2021 (UTC)
0 1 2 0 1, 1 2, 2 0 2 0 1
a a a b a b -> c c b c
For the regular Polyhedra (with n sides), I've managed to work out that the faces can *always* be split into q equal connected pieces if q divides n *except* for 4 and 5 face pieces on the icosahedra. Can someone help me figure those out (or let me know that there is no such?) For example, a Dodecahedron can be covered by 6 identical 2 pentagon units, 4 identical 3 pentagon units, 3 identical 4 pentagon units and 2 identical 6 pentagon units. Naraht ( talk) 18:43, 28 June 2021 (UTC)